The first term in Eq. (REF ) represents the change of the energy of \(\mathcal {S}\)  due to the interaction with the reservoirs \(\mathcal {E}_{\alpha }\) , and can be associated with the total heat exchanged between \(\mathcal {S}\)  and \(\mathcal {E}\) , which we denote as \(\dot{Q}_{\mathcal {S}}\) . The second term in Eq. (REF ) accounts for the variation of the energy of \(\mathcal {S}\)  due to the explicit time dependence of \(H_{\mathcal {S}}\) , and can be associated with the power \(\dot{W}_{\mathcal {S}}\)  injected into \(\mathcal {S}\)  by the driving field. Thus, using these definitions Eq. (REF ) reads as: \( d \langle H_{\mathcal {S}} \rangle / dt = \dot{Q}_{\mathcal {S}} + \dot{W}_{\mathcal {S}}\) . In Ref. [1]} it was shown that the coupling with the reservoirs can induce a stable stationary regime where the state of \(\mathcal {S}\)  is \(\tau _{d}\) -periodic (see the Suplemental Material (SM) for more details). In this regime the average of \(d \langle H_{\mathcal {S}} \rangle / dt\)  over a period of the driving vanishes. This implies that the average power \(\dot{\bar{W}}_{\mathcal {S}}\)  and the average heat current \(\dot{\bar{Q}}_{\mathcal {S}}\)  satisfy that \(0 = \dot{\bar{Q}}_{\mathcal {S}} + \dot{\bar{W}}_{\mathcal {S}}\) . Moreover, \(\dot{\bar{Q}}_{\mathcal {S}}\)  is determined by the variation of the energy of the reservoirs: \(\dot{\bar{Q}}_{\mathcal {S}} = - \dot{\bar{Q}}_{\mathcal {E}}\) , where \(\dot{\bar{Q}}_{\mathcal {E}}\)  is the average of \(\dot{Q}_{\mathcal {E}} = d \langle H_{\mathcal {E}} \rangle / dt \) . To simplify our notation, from now on we will use \(\Delta \mathcal {Q}\)  for the total heat exchanged between \(\mathcal {E}\)  and \(\mathcal {S}\) , i.e. \(\Delta \mathcal {Q} \equiv \tau _{d} \dot{\bar{Q}}_{\mathcal {E}}\) . Considering this, the time average of Eq. (REF ) implies that \(0 = \dot{\bar{W}}_{\mathcal {S}} - \Delta \mathcal {Q} / \tau _{d}\) . In what follows we will compute \(\Delta \mathcal {Q}\)  and use the previous equation to obtain the extracted work and study the efficiency of the engine.
